Short rate ($r_t=f(t,t)$)

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.

The Gaussian Hull-White model has pricing dynamics $d{r}_t=(\beta (t)-a{r}_t)dt+\eta d{W}_t^Q$. Its deterministic drift function can fit the initial yield curve exactly, while its Ornstein-Uhlenbeck process part controls stochastic variation. Like the Vasicek model, it permits negative rates.

Let $f_0(t)$ be the initial instantaneous forward rate curve and choose $r_0=f_0(0)$. In the Hull-White model, take $\beta (t)=f_0^{\prime }(t)+a{f}_0(t)+{\eta }^2(1-e^{-2at})/(2a)$. To verify the fit, write $r_t=x_t+\varphi (t)$ with $d{x}_t=-a{x}_{t}dt+\eta d{W}_t^Q$, $x_0=0$, and $\varphi (t)=f_0(t)+{\eta }^2(1-e^{-at}{)}^2/(2a^2)$. The Gaussian exponential formula gives $-{\partial }_T\log P(0,T)=\varphi (T)-{\eta }^2(1-e^{-aT}{)}^2/(2a^2)=f_0(T)$.

A one-factor short-rate model specifies $d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$ with deterministic coefficient functions. All zero-coupon bond prices are functions of time, maturity and this single scalar state. The physical drift and pricing drift differ by $\eta \lambda$, where $\lambda$ is the market price of risk.

For smooth derivative prices depending on a Markov short rate, Itô formula gives physical drift $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and exposure $s=\sigma V_r$. The common short-rate market price of risk gives $m-rV=\theta s$, yielding this equation. Under the risk-neutral measure, the rate drift is $b-\sigma \theta$ and the bank-discounted price is a martingale under suitable integrability. A unit zero-coupon bond has terminal value one and price ${\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$.

If all traded zero-coupon bonds depend on one scalar rate driven by one Brownian motion, their diffusion exposures form one vector $v_i=\eta P_{i,r}$. Their instantaneous covariance matrix is $v{v}^T$, of rank at most one. Nondegenerate maturity changes are therefore instantaneously perfectly correlated, with correlation $+1$ or $-1$, though finite-horizon bond-return correlations need not be one.

For pricing drift ${\mu }_Q$, a unit zero-coupon bond price $P(t,r;T)$ satisfies $P_t+{\mu }_Q{P}_r+{\eta }^2{P}_{rr}/2-rP=0$, with $P(T,r;T)=1$. Itô formula makes its bank-account-discounted value a local martingale; suitable integrability gives $P(t,r;T)={\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$. The rate kills the diffusion semigroup, as in the Feynman-Kac formula.

In a one-factor short-rate model, a derivative $V(t,r)$ can be hedged locally by a nondegenerate traded zero-coupon bond $P(t,r;T_1)$. Match Brownian exposures by holding $V_r/P_r$ bonds and putting the remaining value into the bank account. Substitution into the two pricing equations proves the gain identity. This requires $\eta P_r\ne 0$; completeness at a degenerate boundary cannot be inferred from the dimension count alone.

When ${\mu }_Q=a(b-r)$ and $\eta (r{)}^2={\eta }_0^2+{\eta }_1^{2}r$, substituting $P=A(\tau )e^{-B(\tau )r}$ gives $B^{\prime }=1-aB-{\eta }_1^2{B}^2/2$ and $(\log A{)}^{\prime }=-abB+{\eta }_0^2{B}^2/2$, with $B(0)=0$, $A(0)=1$. The Vasicek model uses ${\eta }_1=0$, and the CIR model uses ${\eta }_0=0$.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.

Under a risk-neutral measure $Q$, let the short rate be $r_t=g(t)+\sigma W_t^Q$, with deterministic locally integrable $g$ and constant $\sigma >0$. For a zero-coupon bond paying one at $T$, the risk-neutral pricing value isIndeed, conditional on ${\mathcal{F}}_t$, the random part of the integrated future short rate is $\sigma {\int }_t^T(T-u)d{W}_u^Q$, which is Gaussian with mean zero and variance ${\sigma }^2(T-t{)}^3/3$. Its exponential moment gives the formula. At points where $g$ is continuous, the instantaneous forward rate isFor merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative short rates and gives explicit bond prices and forward curves.

To match a prescribed initial instantaneous forward rate curve $F_0(T)$ in the Gaussian short-rate model with a deterministic shift, chooseSince $W_0^Q=0$, its model forward rate is then $f_0(T)=g(T)-{\sigma }^2{T}^2/2=F_0(T)$. Moreover,so the initial zero-coupon bond curve matches as well. A continuous initial forward curve gives pointwise calibration; a locally integrable curve gives the corresponding almost-everywhere interpretation of the instantaneous forward rate.

For $d{r}_t=a_{0}dt+b_{0}d{W}_t$ under a risk-neutral measure, the instantaneous forward rate and zero-coupon bond prices areIndeed, conditional on current information, ${\int }_t^T{r}_{s}ds$ is Gaussian with mean $(T-t)r_t+a_0(T-t{)}^2/2$ and variance $b_0^2(T-t{)}^3/3$. Its negative exponential expectation gives the bond formula, and differentiating its logarithm gives the instantaneous forward rate. Equivalently the affine forward-rate equation has $A^{\prime }=0$, $A(0)=1$, and $B^{\prime }=a_0-b_0^2\theta$, $B(0)=0$. This is the constant-drift instance of the Ho-Lee Gaussian short rate model; the model permits negative short rates.

A nonnegative mean-reverting short-rate diffusion with positive parameters. Its mean-reversion level is $a/b$ and its diffusion size is proportional to $\sqrt{r}$. The Feller positivity condition for the CIR model controls whether zero is reachable. Its stationary law is a gamma distribution and its zero-coupon bond prices are exponential-affine.

For $d{C}_t=a(b-C_t)dt+\sigma \sqrt{C_t}d{W}_t$, the Itô formula applied to $Z=\sqrt{C}$ gives the displayed dynamics before hitting zero. In particular $2ab={\sigma }^2$ does not eliminate the reciprocal drift. The cancellation condition is $4ab={\sigma }^2$. Under that condition $C$ can be represented as the square of a signed Ornstein-Uhlenbeck process with drift $-aY/2$ and volatility $\sigma /2$; its principal square root is the reflected process $|Y|$, not an unrestricted signed OU process. The mean of $C$ is $b+(C_0-b)e^{-at}$ for all admissible positive parameters.

For the CIR model under its money-market risk-neutral measure, the Feynman-Kac formula gives $B^{\prime }=1-bB-{\sigma }^2{B}^2/2$ and $A^{\prime }/A=-aB$, with $B(0)=0$, $A(0)=1$. Solving this Riccati equation gives explicit exponential-affine zero-coupon bond prices. The positive root parameter is $\sqrt{b^2+2{\sigma }^2}$; the squared diffusion coefficient, not the volatility itself, enters this expression.

For a CIR model started at a positive rate, this condition makes the zero boundary inaccessible. Below the threshold, zero can be reached, but the usual nonnegative solution does not cross into negative rates. Thus nonnegativity and strict positivity are different properties.